Find the integral of xlogx/(1+x^2)^2 under the limits 0 to infinity.
\(int frac{xln left(xright)}{left(1+x^2right)^2}dx\)
\(frac{xln left(xright)}{left(1+x^2right)^2}=left(xln left(xright)right)frac{1}{left(1+x^2right)^2}\)
\(=int :xln left(xright)frac{1}{left(1+x^2right)^2}dx\)
\(mathrm{Apply:Integration:By:Parts:}:u=ln left(xright),:v’=frac{1}{left(1+x^2right)^2}x\)
\(=-frac{ln left(xright)}{2left(1+x^2right)}-int :-frac{1}{2xleft(1+x^2right)}dx\)
\(int :-frac{1}{2xleft(1+x^2right)}dx=-frac{1}{2}left(ln left|xright|-frac{1}{2}ln left|x^2+1right|right)\)
\(=-frac{ln left(xright)}{2left(1+x^2right)}-left(-frac{1}{2}left(ln left|xright|-frac{1}{2}ln left|x^2+1right|right)right)\)
\(=-frac{ln left(xright)}{2left(1+x^2right)}+frac{1}{2}left(ln left|xright|-frac{1}{2}ln left|x^2+1right|right)\)
\(=-frac{ln left(xright)}{2left(1+x^2right)}+frac{1}{2}left(ln left|xright|-frac{1}{2}ln left|x^2+1right|right)+C\)
On computing the boundaries,
=0-0
0
